A Gauss-Kuzmin Theorem for Continued Fractions Associated with Nonpositive Integer Powers of an Integerm≥2

We consider a family{τm:m≥2}of interval maps which are generalizations of the Gauss transformation. For the continued fraction expansion arising fromτm, we solve a Gauss-Kuzmin-type problem.

A New Approach for GNSS Ambiguity Decorrelation

Integer carrier phase ambiguity resolution is the key to fast and high-precision Global navigation satellite system(GNSS) positioning and application. LAMBDA method is one of the best methods for fixing integer ambiguity. The principle of LAMBDA is discussed. For incompleteness of Cholesky decomposition and complexity of Integer Gauss transformation, a new approach for GNSS ambiguity decorrelation is proposed based on symmetric pivoting strategy and united inverse integer strategy. The new algorithm applies symmetric pivoting strategy to ambiguity covariance matrix while doing Cholesky decomposition, then finds the inverse and integer matrix of ‘L’. This method not only uses Cholesky decomposition to improve efficiency, but also avoids complicated Integer Gauss transformations. The feasibility and advantage of the method are verified using randomly simulation covariance matrix.

A Gauss-Kuzmin-Lévy theorem for a certain continued fraction

We consider an interval map which is a generalization of the well-known Gauss transformation. In particular, we prove a result concerning the asymptotic behavior of the distribution functions of this map.

On pivot rules for simplex method of interior points, and their investigation on Klee-Minty cube

In [25] it was proposed a parametric linear transformation, which is a "convex" combination of the Gauss transformation of elimination method and the Gram-Schmidt transformation of modified orthogonalization process. Using this transformation, in particular, elimination methods were generalized, Dantzig's optimality criterion and simplex method were developed [26]. The essence of the simplex method development is the following. At each sth step the pivot (positive) vector of length Ks is selected, that allows us to move to improved feasible solution after the step of the generalized Gauss-Jordan complete elimination method. In this method the movement to the optimal point takes place over pseudobases, i.e., over interior points. This method is parametric and finite. Since the method is parametric there are various variants to choose the pivot vectors (rules), in the sense of their lengths and indices. In this article we propose three rules, which are the development of Dantzig's first rule. These rules are investigated on the Klee-Minty cube (problem) [14, 31]. It is shown that for two rules the number of steps necessary equals to 2n, and for third rule we obtain the standard simplex method with the largest coefficient rule, i.e., the number of steps for solving this problem is 2n - 1.